Functional limit theorems for quasi-stationary Hawkes processes
In this talk, we introduce several functional law of large numbers (FLLN) and functional central limit theorem (FCLT) for quasi-stationary Hawkes processes. Under some divergence conditions on triggered events, We prove that the normalized point processes can be approximated in distribution by a long-range dependent Gaussian process. Differently, both FLLN and FCLT fail when triggered events satisfy aggregation conditions. In this case, we prove that the rescale Hawkes process converges weakly to the integral of a critical branching diffusion with immigration. Also, the convergence rate is deduced in terms of Fourier-Laplace distance bound and Wasserstein distance bound. This talk is based on a joint work with Ulrich Horst.
Utility indifference pricing with high risk aversion and small linear price impact
We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.
A Tikhonov Theorem for McKean Vlasov SDEs and an application to mean-field control problems.
We present a stochastic Tikhonov theorem for two-scales systems of SDEs, which cover the case of McKean-Vlasov SDEs. Our approach extends and generalizes previous results on two-scales systems of SDEs without mean-field interaction. As an application we provide a novel method for approximating the solution of certain systems of FBSDEs, related to the Pontryagin maximum principle, which is new even for the case without mean-field interaction. This is a joint work with A. Cosso.